# A geometric rank of Riemannian manifolds

There are various ranks which have been assigned to a smooth manifold. The following ranks are two examples of such ranks:

1. The maximum number of global independent vector fields which can be defined on the manifold
2. The maximum number of independent commuting vector fields on a manifold

In this question, we introduce a geometric rank as follows:

Let $$(M,g)$$ be a Riemannian manifold with a corresponding curvature tensor $$R$$.

The geometric rank of $$M$$, denoted by $${\rm Gr}(M)$$ is defined as the maximum number of independent vector fields on $$M$$ which mutually generate a plane with zero sectional curvature.

Question: What is this geometric rank for $$S^3$$ or $$S^7$$ with their standard structures (standard smooth structures and standard Riemannian metric)?

• If „Standard Riemannian metric“ means the one of constant curvature, then obviously no plane has zero sectional curvature. – ThiKu Nov 4 '18 at 17:26
• The term corank might be more for this concept, since (at least if we replace $M$ with a suitable open subset) it is maximal for a flat metric. This local version should be related to the rank of $R$ viewed as an element of $\operatorname{End} \bigwedge^2 TM$. – Travis Nov 4 '18 at 17:49
• @Travis The rank of a symmetric space (or more generally, of a manifold with either $\mathrm{sec}\le 0$ or $\mathrm{sec}\ge 0$) is the maximal dimension of a complete immersed totally geodesic flat manifold. So, ranks in geometry tend to refer to ranks of abelian groups rather than to ranks of endomorphisms. On the other hand, I would not know how the rank I alluded to relates to the one presented here. – Sebastian Goette Nov 10 '18 at 10:38